Surely you can find various ways to reason or calculate these unit conversions.
But for practical purposes adding a half goes a long way. And it's even easier to add the missing 0.1, if you really need to.
A mathematician's way of converting miles to kilometers
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Re: A mathematician's way of converting miles to kilometers
#202I've been doing it like that as well for years. Most people you explain it (replace the number with a sum of fibonacci numbers, and for each one, take the next) to come back with "but it becomes less accurate for larger numbers, right?". After you say "hm, no!" there's a pause, and then the penny drops: Golden ratio! Anyway, I can't remember where I learned it.
It's better for large numbers since the asymptotic property dominates. It's questionable for small numbers since then the effect of the initial condition dominates. For example, the tweet's argument works the same way for the Fibonacci sequence that goes 1,3,4,7,11,... but obviously that gives different numbers.
By the way, those are called the Lucas numbers.
Re: A mathematician's way of converting miles to kilometers
#203Earlier quoted context omitted.
You are correct that "10" is a very special number, as long as you don't assume that it can only mean "ten". In fact, every number base is base "10" when you interpret the "10" in that base . Try it: 10 binary is two. 10 octal is eight. 10 hexadecimal is sixteen. This is the very definition of a number base: it is the multiplier that you represent by appending "0" to a string of numeric characters in that base. So th…
This illustrates the idea well: https://i.imgur.com/II5W6Pl.png
Amusingly, the alien and astronaut would be referring to the same thing when they say 'Base 3'
Re: A mathematician's way of converting miles to kilometers
#204Division and multiplication by 5 and 8 should be fairly easy for most folks, imho. Lot easier than trying to remember the closest Fibonacci number to me. Quick, what’s the closest Fibonacci number to 150? Can you do that faster than 150/5*8 in your head? What about 500? The reverse is almost as easy. Even with numbers not as evenly divisible, say 490km, most will know 490/8 is about 61 quickly. Multiple that by 5 and…
I'm admittedly pretty bad at basic speed math (even dividing by 5 takes me a bit -- and I know it shouldn't). I use the same fibonacci trick, and at least for me it really is much easier. It's important to remember that all arithemtic tricks are made more useful when combined with others. I don't know what fibonacci number is close to 500, but I don't need to: 5 -> 8 means 500 -> 800. Really, the only fibonaccis I ha…
> 150 is harder, but I would use the same trick.
150/5 is something literally any adult should be able to do instantly. I realize that’s a bit hyperbolic, but still seriously easy. 30*8 is also very simple too.
So yeah, I get what you’re saying, but seriously, practice a little and I swear you’ll be able to learn it.
Re: A mathematician's way of converting miles to kilometers
#205Surely you can find various ways to reason or calculate these unit conversions. But for practical purposes adding a half goes a long way. And it's even easier to add the missing 0.1, if you really need to.
When I'm in the US I find that with a little practice I can guess it with surprising accuracy. It might help that I have mild synaesthesia so visualising a number line makes it possible to almost see the conversion.
Re: A mathematician's way of converting miles to kilometers
#206I basically just use 6. 1.6 is the factor everyone talks about (approximation of 1.609), but it has 2 significant figures. To make mental calculations easier/quicker I use its reciprocal, 0.621, which I approximate with 0.6, which has only one significant figure, 6. Instead of multiplying by 1.6 you would divide by 0.6, which basically amounts to dividing by 6 and then moving the decimal point to someplace plausible.…
> 1.6 is the factor everyone talks about (approximation of 1.609), but it has 2 significant figures. To make mental calculations easier/quicker I use its reciprocal, 0.621, which I approximate with 0.6, which has only one significant figure, 6. How about just 1.5 + 0.1? I.e. the same amount plus half, plus a tenth? So, 55 mi -> 55 + (25 + 2.5) + 5.5 -> 88km And inversely, 0.5 + 0.1, so half plus a tenth: 80km -> 40 +…
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Re: A mathematician's way of converting miles to kilometers
#207Earlier quoted context omitted.
There's an argument against intelligent design right there (4 or 6 fingers per hand are obviously better).
Eh, evolution is a pretty nifty mix of oo class extensions, recursion, brute-force and bias weightings. I'd wager Gawsh made the best system S/He could given product constraints (completely unfocused if you ask me [which I know no one did]) and the real need to deliver (take it easy over there Leibniz, the world is still crap as evidenced everywhere). Anyway, can't knock it 'til you've built it. This is an interestin…
Well, my comment was partly in jest (though I do think it's by no means clear that 5 is a local optimum, thus it's quite possible that 4 or 6 would be better, and twice either would give us a better base for counting), but I'm amazed that there's actual scientific discussion of the issue. I wish to quote the most pertinent part of the article though:
> Is there really any good evidence that five, rather than, say, four or six, digits was biomechanically preferable for the common ancestor of modern tetrapods? The answer has to be "No,"
Re: A mathematician's way of converting miles to kilometers
#208Earlier quoted context omitted.
Physicists have their own battles, like electrical engineers using Ohm's Law as a definition of impedance.
What's the battle there? How do physicists define impedance?
Ohm's law is an empirical law that only holds in certain circumstances. A classic exercise is measuring the current and voltage across a lightbulb, plotting it, and measuring the slope of the line. The slope is the impedance. Then you turn up the voltage and watch the line turn into a curve, which is where the law breaks down and doesn't apply anymore. The engineers treat it like a definition and assume linearity over all voltage.